(0) Obligation:
The Runtime Complexity (innermost) of the given
CpxTRS could be proven to be
BOUNDS(1, 1).
The TRS R consists of the following rules:
f(f(x, y, z), u, f(x, y, v)) → f(x, y, f(z, u, v))
f(x, y, y) → y
f(x, y, g(y)) → x
f(x, x, y) → x
f(g(x), x, y) → y
Rewrite Strategy: INNERMOST
(1) DependencyGraphProof (BOTH BOUNDS(ID, ID) transformation)
The following rules are not reachable from basic terms in the dependency graph and can be removed:
f(f(x, y, z), u, f(x, y, v)) → f(x, y, f(z, u, v))
(2) Obligation:
The Runtime Complexity (innermost) of the given
CpxTRS could be proven to be
BOUNDS(1, 1).
The TRS R consists of the following rules:
f(x, x, y) → x
f(x, y, y) → y
f(x, y, g(y)) → x
f(g(x), x, y) → y
Rewrite Strategy: INNERMOST
(3) CpxTrsToCdtProof (BOTH BOUNDS(ID, ID) transformation)
Converted Cpx (relative) TRS to CDT
(4) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(z0, z0, z1) → z0
f(z0, z1, z1) → z1
f(z0, z1, g(z1)) → z0
f(g(z0), z0, z1) → z1
Tuples:
F(z0, z0, z1) → c
F(z0, z1, z1) → c1
F(z0, z1, g(z1)) → c2
F(g(z0), z0, z1) → c3
S tuples:
F(z0, z0, z1) → c
F(z0, z1, z1) → c1
F(z0, z1, g(z1)) → c2
F(g(z0), z0, z1) → c3
K tuples:none
Defined Rule Symbols:
f
Defined Pair Symbols:
F
Compound Symbols:
c, c1, c2, c3
(5) CdtLeafRemovalProof (BOTH BOUNDS(ID, ID) transformation)
Removed 4 trailing nodes:
F(z0, z0, z1) → c
F(z0, z1, z1) → c1
F(z0, z1, g(z1)) → c2
F(g(z0), z0, z1) → c3
(6) Obligation:
Complexity Dependency Tuples Problem
Rules:
f(z0, z0, z1) → z0
f(z0, z1, z1) → z1
f(z0, z1, g(z1)) → z0
f(g(z0), z0, z1) → z1
Tuples:none
S tuples:none
K tuples:none
Defined Rule Symbols:
f
Defined Pair Symbols:none
Compound Symbols:none
(7) SIsEmptyProof (BOTH BOUNDS(ID, ID) transformation)
The set S is empty
(8) BOUNDS(1, 1)